21. Cylindrical Symmetry: Bessel Functions
The Bessel functions are solutions to Laplace’s equation in cylindrical coordinates that is:
The usual separation of variables routine leads to solutions
The first two equations are of course simple: For positive the solutions for will be exponentials.
For negative is oscillatory, etc.
To ensure a single-valued potential, the equation for the angular variation, , must have solutions with an integer.
For the positive case (often called hyperbolic behavior in the direction) the change of variable puts the radial equation into what is the standard Bessel form:
In contrast to the Legendre polynomials for the spherical case, these Bessel functions are infinite series.
The Bessel equation is second-order, so for given there are two independent solutions.
First look at the equation near the origin: the 1 in the term can be dropped in the limit and the solutions are then proportional to Now look far away: the is negligible, and solutions look like combinations of
(By the way, this amplitude is just what we need for propagating waves in two dimensions, discussed next semester, because the wave energy goes as the square of the amplitude, so as the wave goes outwards the same total energy crosses successive circles (or cylindrical surfaces) of circumference )
Here are plots of those solutions which are well-behaved at the origin, with the standard normalization.
Behavior near the origin and far away:
These are called cylindrical Bessel functions.
The behavior goes from power law to oscillation in the region
In fact, it's not difficult to find the exact solution as a series. Trying a solution in the differential equation, we find , coefficients of odd powers are zero, and successive coefficients of even powers have simple ratios, the result for the nonsingular solution is
The Neumann Function
(Graph from Wikipedia Commons)
For integer , so we have only found one solution. The other is called the Neumann function, defined as
it remains independent even as becomes integral (details in Stone and Goldbart, p 279).
The Neumann function is singular at the origin:
but similar to far away:
The transition from small to large behavior again takes place when
Hankel Functions (in and out waves at infinity)
The Bessel functions of the third kind, called Hankel functions, are the linear combinations:
Evidently, these behave at large as outgoing and incoming waves:
Modified Bessel Functions (exponential at infinity)
If in solving in a cylinder the given boundary conditions require oscillatory behavior in the direction, then and the Bessel functions have pure imaginary argument. It is convenient to introduce new notation in this case, so we define
real functions for real
Exercise: check the behavior of these functions for very small and very large argument by substituting in the results above for Bessel and Hankel functions. In particular, note that has exponential increase, exponential decrease at large apart from the usual term.
Radial Equation Inside a Grounded Cylinder
Since the interior of the cylinder includes the radial origin, this solution must be made up of
The will in fact be the integer from the azimuthal equation.
Furthermore, the solution is identically zero on the cylinder so the only possibilities are the where and the are the zeroes of the Bessel function,
These zeroes are of course tabulated, and the first few are given in Jackson: the spacing soon tends to :
You can see these, approximately, on the graphs of the first 's given above.
The Fourier-Bessel Series
Suppose now we have to solve inside a cylinder, going to zero at the walls
We shall show that the set of functions are an orthogonal set on An expansion of an arbitrary function in terms of these is called a Fourier-Bessel series.
The equation satisfied by is:
Take any to be any of the remember the are the zeroes of so
Now multiply the first equation by , the second by and subtract to get
Integrating from to the term on the right is zero at both ends, so
Notice that this integral includes a weight function . To construct an orthonormal basis, we need also that
(Proving this requires recursion properties of the Bessel functions we'll discuss later.)
We can now expand an arbitrary function equal to zero at on the interval :
This is the conventional Fourier-Bessel series, for functions vanishing at that is, Dirichlet boundary conditions.
Another possible expansion is in functions whose derivatives vanish at the boundary, that is, where
But there’s more: if we replaced by the would be or The Bessel equation would be changed to
and the solutions are called modified Bessel functions. These are exponentially growing or decaying functions, they’re really just Bessel functions with pure imaginary argument,
(Graph from Wikipedia Commons)
So what do the possible solutions of inside the cylinder look like? Obviously, that depends on the boundary conditions, but we can make one general remark. Recall from Earnshaw’s theorem that cannot have a minimum in the volume. This means that all three components cannot be of the oscillatory type. The component must be oscillating (or constant) so the other two must be of opposite types. We can see this: if the direction behavior is hyperbolic, the radial is oscillating; if the opposite is the case.
Bessel Generating Function and Identities
The two-dimensional Helmholtz (wave) equation has (nonsingular) solutions Any other nonsingular solution to the wave equation should be expandable in terms of these functions, in particular a plane wave:
In fact we'll show the are all equal to one, so
From this, we have
Tip: any time you come across an exponential of a trig function, you're probably dealing with a Bessel function!
The generating function is also often written (putting )
Expanding the left-hand side and collecting powers of we eventually find
the correct expression for the Bessel function, proving that the coefficients were indeed all equal to one, as assumed. (OK, it was hindsight.)
From the expression for and from the generating function, it's not difficult to find the recurrence relations
Exercise: do it! They both come from differentiation … but with respect to what?